Finding the function of a sine graph that has both translation and transformation

Question

I can't quite find a problem similar enough to this yet, and I need some serious help. Here is a photo of the graph of the function I am trying to find out:

Sorry, but I don't have enough reputation to post a physical copy of the photo yet. I've figured out most of the problem. The basic formula for a sine graph is $\sin(2\pi/P(x−b))$. So far from the picture, I've figured out that the amplitude is $4$ and the frequency (is that right?) is $2\pi/5$ and that $x$ will most definitely be negative. My problem is in finding b, which for the life of me I can't figure out. So far I've hit $4\sin(2\pi/5(-x-1.5))$, but that doesn't seem to be correct. Can you guys explain what I'm doing wrong, so that I can learn how to do this correctly? Thank You

Answer 1

Amplitude and wavelegth(time-period) are ok. You have to recognise two things in addition. The y- shift up and the associated phase angle for x- shift. Try $$ y= 3+ 4 \cos \frac {2 \pi ( x+2)}{ 5} $$

Answer 2

My previous answer was unfortunate--the phase shift, as you noted, is the real challenge here (I'll chalk up my careless previous answer to this being way past my bedtime). Try graphing the following: $4\sin\left[\dfrac{2\pi}{5}\left(x-1-\dfrac{5\sin^{-1}(\frac{3}{4})}{2\pi}\right)\right]+3$:

Given the construction of the function, I imagine you will be able to tell what is going on. If not, feel free to comment.